Okay I have come across this before and there has been a few of these posted on various boards over the years. I don't know exactly what is happening, scientifically, but the basic issue is that there is moisture under the stone, which is not entirely unusual. However when something is placed on top of it, like a mat or a towel, it slows down or completely inhibits the stone's ability to transmit moisture. It's the same color of the stains you see where a potted plant has sat for years, or other standing water. It is basically moisture damage, but without any physical deterioration.
There is also an orange type of mildew/mold like you see in showers sometimes.
I don't know much about it, all I know is I have seen this several times before, it was caused by moisture being trapped under a rubber or thick mat, the stain was limited to the exact area that was covered up, and it couldn't be removed.
At this point, I might try what Robert Falzone suggested, with the explanation to your customer that you're fairly confident you won't be able to get it out, but you;'re willing to try. I honestly think replacing the tiles will be less labor intensive, and the most cost effective approach to eliminating this.
You can do a test with a non acidic rust remover, like Alpha RSR 2000. Non-acidic rust removers turn rust purple on contact. This will confirm whether or not it is rust. If not, you can move on to peroxide or bleach type cleaners without the risk of making it worse.